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M2.Lecture 6 : Two-Way ANOVA Notes (Evidence Based Practice)

Two-Way ANOVA: Concept and Terminology

  • Factorial Analysis of Variance (ANOVA)
  • Simultaneously compare the effect of more than one independent variable on a dependent variable
  • Assumptions:
    • Independence of observations
    • Normality within groups
    • Homogeneity of variances across groups
  • Dependent variable (DV) level of measurement: interval or ratio
  • Key outcomes:
    • Main effects: effect of IV A on the DV; effect of IV B on the DV
    • Interaction effect: combined effect of IV A and IV B on the DV

Two-Way ANOVA: Main Effects and Interaction

  • Main Effects:
    • What is the effect of IV A on the DV?
    • What is the effect of IV B on the DV?
  • Interaction Effect:
    • What is the combined effect of IV A and IV B on the DV?
    • Interaction means the effect of one IV depends on the level of the other IV

Hypothesis Testing Framework (Hypothesis Testing Review)

  • General steps:
    • 1) Develop null and research hypotheses
    • 2) Choose a level of significance
    • 3) Determine which statistical test is appropriate
    • 4) Run analysis to obtain test statistic and p value
    • 5) Make a decision about rejecting or failing to reject the null hypothesis
    • 6) Make a conclusion
  • For a Two-Way ANOVA with the politics interest example, the null and research hypotheses are:
    • H0_1: There will be no difference in political interest between male and female students
    • H1_1: There will be a difference in political interest between male and female students
    • H0_2: There will be no difference in political interest among students of different education levels
    • H1_2: At least one education level group will have a different level of political interest than the others
    • H0_3: There will be no interaction between sex and education level
    • H1_3: There will be an interaction between sex and education level

Significance Level and Test

  • Level of significance: \alpha = 0.05
  • Test: Factorial (Two-Way) ANOVA
  • Assumptions reiterated:
    • Independence
    • Normality
    • Homogeneity of variances
    • DV measured at interval/ratio

Example Study Design (Politics Interest)

  • Objective: Examine differences in reported interest in politics by gender (male, female) and by education level (school, college, university)
  • DV: Interest in politics
  • Design: Two IVs with 2 and 3 levels respectively; looks at main effects and interaction

Reported Results: Hypothesis Tests (as in transcript)

  • Gender main effect:
    • F(1,54) = 1.628,\; p = 0.207
    • Interpretation: p > 0.05, fail to reject H0 for gender; no statistically significant difference between male and female students on political interest
  • Education main effect:
    • F(2,54) = 147.517,\; p < 0.001
    • Interpretation: reject H0 for education; there is a statistically significant difference among education level groups on political interest
  • Gender by Education interaction:
    • F(2,54) = 4.643,\; p = 0.014
    • Interpretation: reject H0 for interaction; there is a statistically significant interaction between gender and education level on political interest

Additional Results and Interpretation

  • A two-way ANOVA indicated significant interaction between gender and education level on political interest, F(2,54) = 4.643, p = 0.014
  • The education main effect was highly significant, F(2,54) = 147.517, p < 0.001
  • The gender main effect was not significant, F(1,54) = 1.628, p = 0.207
  • Simple main effects analysis indicated that interest in politics differed significantly among education levels (p < 0.001)
  • Summary interpretation:
    • There is no overall difference by gender when averaging across education levels
    • Differences exist across education levels, but the presence of a significant interaction means the effect of education on interest depends on gender
    • Because the interaction is significant, main effects should be interpreted with caution; explore simple main effects and plot interaction patterns

Practical Implications of the Interaction

  • A significant interaction implies that the impact of education level on political interest varies by gender (and possibly vice versa)
  • When interaction is present, focus on simple main effects to identify where differences lie
  • Large education effect suggests meaningful differences across education levels, but the interpretation must account for the interaction

Simple Main Effects and Post-Hoc Note

  • The transcript notes a simple main effects result: interest in politics differed significantly among education levels (p < 0.0001, i.e., p < 0.001)
  • This supports the finding of differences across school, college, and university levels

Formulas and Notation (LaTeX)

  • General ANOVA F-statistic: F = \frac{MS{\text{Between}}}{MS{\text{Within}}}
  • For two-way ANOVA, separate F-tests:
    • Main effect A: FA = \frac{MSA}{MS_{\text{Error}}}
    • Main effect B: FB = \frac{MSB}{MS_{\text{Error}}}
    • Interaction AB: F{AB} = \frac{MS{AB}}{MS_{\text{Error}}}
  • Degrees of freedom (typical):
    • df_A = a - 1
    • df_B = b - 1
    • df_{AB} = (a-1)(b-1)
    • df_{\text{Error}} = N - ab
    • df_{\text{Total}} = N - 1
  • P-value interpretation is the same: if p < α, reject the null hypothesis; otherwise fail to reject

Hypothesis Testing Process: Condensed

  • Step 1: Develop null and research hypotheses for each effect (A, B, and AB)
  • Step 2: Choose α = 0.05
  • Step 3: Determine the statistical test (factorial two-way ANOVA)
  • Step 4: Run the analysis to obtain test statistics and p-values
  • Step 5: Decide to reject or fail to reject each null hypothesis
  • Step 6: Draw conclusions in the context of the study

Connections and Context

  • Relation to one-way ANOVA and to regression concepts
  • Two-way design extends to more factors in a factorial setup
  • Understanding main effects vs interaction is essential for correct interpretation
  • Real-world relevance: many studies involve demographic and educational variables that interact to influence outcomes
  • Ethical and practical implications: avoid overgeneralization, consider sample size and power, ensure fair interpretation across groups

Quick Reference Summary

  • Two-Way ANOVA assesses three effects: A, B, and AB
  • Interpret main effects with caution if a significant interaction is present
  • Use simple main effects to understand where differences lie
  • Report results with F-statistics, degrees of freedom, and p-values, and provide a practical interpretation in the study context